Handbook of Linear Algebra, 2nd Edition

Chapter 1

Vectors, Matrices, and Systems of Linear Equations

Jane Day

San José State University

Throughout this chapter, F will denote a field. The references [Lay12], [Leo10], and [SIF08] are good sources for more detail about much of the material in this chapter. They discuss primarily the field of real numbers, but the proofs are usually valid for any field. Vector spaces over a general field are presented in [FIS03]

1.1 Vector Spaces

Vectors are used in many applications. They often represent quantities that have both direction and magnitude, such as velocity or position relative to the origin. They can appear as functions, as n-tuples of scalars, matrices, or in other disguises. Whenever objects can be added and multiplied by scalars, they ...